MinT - Best Known (255−169, 255, s)-Nets in Base 4

Best Known (255−169, 255, s)-Nets in Base 4

(255−169, 255, 104)-Net over F₄ — Constructive and digital

Digital (86, 255, 104)-net over F₄, using

t-expansion [i] based on digital (73, 255, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(255−169, 255, 129)-Net over F₄ — Digital

Digital (86, 255, 129)-net over F₄, using

t-expansion [i] based on digital (81, 255, 129)-net over F₄, using
- net from sequence [i] based on digital (81, 128)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 81 and N(F) ≥ 129, using
    - Drinfeld modules of rank 1 [i]

(255−169, 255, 640)-Net in Base 4 — Upper bound on s

There is no (86, 255, 641)-net in base 4, because

1 times m-reduction [i] would yield (86, 254, 641)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 916 003928 902262 854160 866480 157871 850227 937240 521896 053857 155533 077453 500085 896075 503417 678355 432028 881819 613268 487334 385474 826720 194831 517860 354618 328992 > 4²⁵⁴ [i]